Problem 41 - Elementary Algebra

Question

For the following problems, expand the quantities so that no exponents appear. $$ 10 a^{3} b^{2}(3 c)^{2} $$

Step-by-Step Solution

Answer

Answer: The expanded expression is \(90a^3b^2c^2\).

Step 1: Expand \((3c)^2\)

To expand \((3c)^2\), we simply need to square both terms within the parentheses: \((3c)^2 = (3^2)(c^2) = 9c^2\).

Step 2: Multiply the expanded terms

Now, we need to multiply the three terms: \(10a^3\), \(b^2\), and \(9c^2\). This can be done by multiplying the coefficients (constant values) and applying the product of powers rule for variables: $$ 10a^3 \cdot b^2 \cdot 9c^2 = (10 \cdot 9)(a^3)(b^2)(c^2) = 90a^3b^2c^2 $$ The final expanded expression is \(90a^3b^2c^2\).

Key Concepts

Exponent Rules

Understanding exponent rules is crucial when working with algebraic expressions that contain powers. Exponents represent repeated multiplication of a base number. When expanding exponents, there are key rules to keep in mind: the Product of Powers rule states that when you multiply two powers with the same base, you can simply add the exponents. For instance, if you have \(a^m \times a^n\), it simplifies to \(a^{m+n}\). If exponents are raised to another power, like \( (a^m)^n \), you multiply the exponents to get \(a^{mn}\).

Another important rule is the Power of a Product rule, which means that when a product is raised to a power, each factor in the product is raised to that power separately. For example, \( (ab)^n = a^n b^n\). These rules streamline the process of expanding and simplifying expressions, as seen in the textbook exercise, saving students from lengthy and unnecessary calculations.

Simplifying Algebraic Expressions

Simplifying algebraic expressions is a fundamental skill in algebra. The process often includes expanding exponents, combining like terms, and reducing fractions. It's essential to follow a systematic approach to avoid errors. Begin by handling the parentheses using the distributive property or applying exponent rules if they contain powers. Next, look for like terms—those that have the same variable parts and exponents—and combine them by adding or subtracting their coefficients.

In the exercise example, simplification is done through exponent expansion and multiplication. Once the expression is expanded, we do not have like terms to combine, as all the variable parts are different. When you encounter expressions with similar variable parts across terms, gathering and simplifying them will further refine the expression to its simplest form.

Product of Powers

The product of powers rule plays a pivotal role when you are multiplying algebraic expressions that contain exponents. This rule, as mentioned before, states that for any base that is multiplied by itself, you add the exponents. However, when different bases are involved, as in our exercise, you multiply the individual terms while keeping the exponents intact.

When working with products that involve coefficients (numerical factors) and variables with exponents, it's best to multiply the coefficients together and then deal with the variables separately. For instance, a scenario such as \(10a^3b^2 \cdot 9c^2\) first leads us to multiply the numbers 10 and 9 to get 90. Then we consider the variable terms. Since the bases \(a\), \(b\), and \(c\) are not the same, we do not use product of powers rule between them; instead, we write them next to each other keeping their respective exponents. Consequently, the expression simplifies to \(90a^3b^2c^2\), as shown in the step-by-step solution.